Question

How do you solve $2x+A=B$ given $A=\left[ \begin{matrix} 2 \\\ 9 \\\ -2 \\\ \end{matrix}\begin{matrix} -8 \\\ 5 \\\ 3 \\\ \end{matrix} \right]$ and $B=\left[ \begin{matrix} -6 \\\ 1 \\\ 8 \\\ \end{matrix}\begin{matrix} 2 \\\ -5 \\\ 5 \\\ \end{matrix} \right]$?

Answer 1

Consider x as a variable matrix and solve for the value of x. Take the matrix A to the R.H.S. and subtract the elements of matrix A from the corresponding elements of matrix B. Now, divide both the sides with 2 to make the coefficient of x equal to 1. Accordingly divide each element of the resultant matrix in the R.H.S. by 2 and get the answer.

Complete step by step solution:
Here, we have been provided with the equation $2x+A=B$ and the values of A and B are given in matrix form. We are asked to solve for the value of x, that means we have to determine the matrix x.
Now, leaving the variable x in the L.H.S. and taking the matrices A and B to the R.H.S., we get,
$\Rightarrow 2x=B-A$
Substituting the given matrix form of A and B, we get,

-6 \\\ 1 \\\ 8 \\\ \end{matrix}\begin{matrix} 2 \\\ -5 \\\ 5 \\\ \end{matrix} \right]-\left[ \begin{matrix} 2 \\\ 9 \\\ -2 \\\ \end{matrix}\begin{matrix} -8 \\\ 5 \\\ 3 \\\ \end{matrix} \right]$$ Here, we need to perform the subtraction operation on two matrices. So, according to the subtraction property of matrices we need to subtract each element of matrix A from the corresponding elements of matrix B. So, we get, $$\begin{aligned} & \Rightarrow 2x=\left[ \begin{matrix} -6-2 \\\ 1-9 \\\ 8-\left( -2 \right) \\\ \end{matrix}\begin{matrix} 2-\left( -8 \right) \\\ -5-5 \\\ 5-3 \\\ \end{matrix} \right] \\\ & \Rightarrow 2x=\left[ \begin{matrix} -8 \\\ -8 \\\ 10 \\\ \end{matrix}\begin{matrix} 10 \\\ -10 \\\ 2 \\\ \end{matrix} \right] \\\ \end{aligned}$$ Dividing both the sides with 2, we get, $$\Rightarrow x=\dfrac{1}{2}\left[ \begin{matrix} -8 \\\ -8 \\\ 10 \\\ \end{matrix}\begin{matrix} 10 \\\ -10 \\\ 2 \\\ \end{matrix} \right]$$ Now, the division or multiplication property in a matrix states that when we multiply or divide a matrix with a scalar then each element of that matrix is to be divided or multiplied with scalar. So, we get, $$\Rightarrow x=\left[ \begin{matrix} -4 \\\ -4 \\\ 5 \\\ \end{matrix}\begin{matrix} 5 \\\ -5 \\\ 1 \\\ \end{matrix} \right]$$ Hence, the above matrix obtained is our answer. **Note:** One may note that here you must not consider the variable x as a simple linear variable. Actually, it is a matrix having six variables which can be represented in matrix form as $$x=\left[ \begin{matrix} {{x}_{1}} \\\ {{x}_{3}} \\\ {{x}_{5}} \\\ \end{matrix}\begin{matrix} {{x}_{2}} \\\ {{x}_{4}} \\\ {{x}_{6}} \\\ \end{matrix} \right]$$. Remember that we can add or subtract two or more matrices only when they have the same order. In the above question the matrices were of the order $$\left( 3\times 2 \right)$$ where ‘3’ represents the number of rows and ‘2’ represents the number of columns. You must remember the division and multiplication property of a scalar with a given matrix, otherwise you will get confused in the last step of the solution.